Opacity is a confidentiality property for systems whose behavior is only partly visible to an outside observer. A system is opaque with respect to a chosen secret if, no matter what that observer sees, it can never become certain the secret occurred. Equivalently: for every run that reveals the secret, there must exist some other run, indistinguishable to the observer, that doesn't.
The notion originates in the discrete-event-systems and labeled-transition-system literature of the mid-2000s, and has since been formalized, verified, and enforced across a wide range of system models: plain finite automata, Petri nets (bounded and unbounded), pushdown automata, timed automata, timed process algebra, and more. The core question stays the same everywhere it goes, but the answer, and the machinery needed to get it, changes considerably depending on what kind of system is being asked about.
The classical setting, and where the field started: finite automata, labeled transition systems, and bounded Petri nets, generally nondeterministic, with a single observer. The largest and most developed page in the index.
02Systems where the observer also reads off when events happen, not just their order, typically via timed automata and real-valued clocks. Timing becomes a side channel in its own right.
03Systems with randomness: probabilistic automata, hidden Markov models, and Markov decision processes. Strict yes/no opacity gives way to quantifying disclosure risk.
04Systems observed by more than one site. The underlying system stays finite-state; what changes is the observation architecture.
05What happens once the finite-state assumption is dropped: pushdown automata, unbounded Petri nets. A much smaller, younger corner of the literature, in part because many natural questions here turn out to be undecidable.
06Cyber-physical and control systems whose state evolves continuously. Exact indistinguishability gives way to a bounded, metric notion of closeness.
07Systems whose transitions carry algebraic weights, typically over the max-plus (tropical) semiring. Information can leak through the weight itself.
08Neighboring notions that keep surfacing while researching opacity: detectability, diagnosability, non-interference, and a few others. This page makes no attempt to be exhaustive about them.
09A small tool over the whole bibliography. Pick tags and pull up every matching paper, across all pages at once.